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Sequences and Series
Sequences Arithmetic Progression tn = a + (n-1)d Sn = n/2 + (n-1)d =n/2 + tn tn = Sn - Sn-1 . . a2 - a1 = an - an-1 = constant a1 + an = a2 + an-1 = constant a1 + k , a2 + k , a3 + k ,......., an + k are also in A.P. a1k , a2k , a3k ,......., ank are also in A.P. a1 , a3 , a5 are also in AP a2 , a4 , a6 are also in AP {terms falling after equal intervals are in AP} . .. 3 consecutive terms : 4 consecutive terms : 5 consecutive terms : AM = (a + b) / 2 ; for two numbers AM = (a1 + a2 + a3 + ..... + an) / n (for n numbers) Geometric Progression tn = arn-1 Sn = a(rn-1) / (r - 1) if r > 1 S∞ = a / (1 - r) if 1 < r tn = Sn /Sn-1 . , a2/a1 = a3/a2 = an/an-1 = common ratio a1an = a2an-1 = constant a1 + k , a2 + k , a3 + k ,......., an + k are also in A.P. a1k , a2k , a3k ,......., ank are also in G.P. a1 , a3 , a5 are also in GP a2 , a4 , a6 are also in GP {terms falling after equal intervals are in GP} . 3 consecutive terms : 4 consecutive terms : 5 consecutive terms : . . GM = √(ab) for two numbers GM = √(a1aca3....an)n (for n numbers) Harmonic Progression tn = 1 / A.P. . . (1 / an) - (1 / an-1) = constant HM = 2ab / (a+b) . 3 consecutive terms : 4 consecutive terms : 5 consecutive terms : Series Mean Arithmetic Mean AM = (a + b)/2 Geometric Mean GM = √(ab) Harmonic Mean HM = 2ab / (a+b) Relation between AM,GM,HM AH = G2 Aritmetico - Geometrico Series a.b + (a+d)br + (a+2d)br2 ........ (a+(n-1)d)brn-1 is called an Arithmetico Geometrico Series For such series , we multiply the equation with r and then subtract it from the original equation . Sum of Finite Series Σn0 = n Σn = n(n-1) / 2 Σn2 = n(n-1)(2n-1) / 6 Σn3 = n(n-1) / 2]2 Note : For other summations , convert the expression into the above given expressions . Infinite Series Sum = a / (1- r) Special Series Special Series works on the Principle of Binomial Expansion . Algebraic Expansions (1-x)-1 = 1 + x + x2 + x3 + ...... ∞ (1+x)-1 = 1 - x + x2 - x3 + ..... ∞ (1 - x)-2 = 1+ 2x + 3x2 + 4x3 + .....∞ (1+ x)-2 = 1 - 2x + 3x2 - 4x3 + ....∞ Exponential Series ex = 1 + x/1! + x2/2! + x3/3!......∞ e-x = 1 - x/1! + x2/2! - x3/3!......∞ Logarithmic Series log(1+x) = x - x2/2 + x3/3 - ....... ∞ log (1-x) = - x - x2/2 - x3/3 - ....... ∞ Tips and Tricks Category:Mathematics